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At the end of The Amazing Spider-Man film, after Dr. Connors is arrested and confined to his asylum cell, he's visited by a mysterious man whose face is never shown and whose name is never given. How the man is able to enter or exit the cell with no one else noticing could be anyone's guess. Dr. Connors does not seem surprised that the man is there.
The man asks Dr. Connors if he told "the boy" the truth about his father. By "the boy," Dr. Connors knows the man means Peter Parker/Spider-Man, and he answers that no, he didn't tell Peter the truth about his late father, Richard. The man then laughs maniacally, says that's a good thing, and leaves. Just as he's leaving, Dr. Connors angrily warns the man to leave Peter alone.
This scene, which we see only during the ending credits, shows that (at least in his human form anyway) Dr. Connors still cares for Peter – and clearly he's not the only one to know something about Peter's father that he's not sharing with him.